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Sound of a guitar string - Vibration frequencies

November , 11th 2018 | Author: Prontubeam (@Prontubeam_en) Read: 901 times

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How many times have you wondered why each guitar string sounds different? There are several reasons, such as the material of which they are made, but it is not the only one. We all know that to tune a guitar we must turn a few pegs located on the top of it (also called peg box). Do we know why we do that? Let’s see it.

When we rotate the peg of the guitar associated with each string, we are subjecting it to an imposed stretching that produces a tension. When we apply this tension, what we are doing is, in reality, changing its rigidity and therefore its frequency of vibration, which is fully related to the sound it emits. We can see on any music website the required vibration of the string to reproduce each particular note.

In this case, let’s suppose we want to make vibrate a steel cable at 450Hz. This problem can be easily addressed through the principle of virtual works (PVW) that is dealt with in specialized literature on the subject (dynamic analysis). In this case we are going to use a calculation program, Ansys, to study the variation of the frequency of a string as a function of the tension to which it is subjected, or said otherwise, depending on the imposed stretching introduced, so we will know how much the peg must be turned.

We provide the input:

Input data

String length: 0.6 meters

String diameter: 0.5 mm

Material’s properties:

                E=2.1e11 (Pa) Young Modulus

                Poisson ratio: 0.3

                Steel density: 7800kg/m3

 

Boundary conditions: One end with all its degrees of freedom blocked the other one pinned

 

Now we are ready to calculate what we are searching for. We begin with the geometry. We model the string with Design Modeler, we convert it to a beam element with Spaceclaim and returning to Design Modeler we give it the section we want (cross section). For the material we make a small change to the steel that comes by default in the library and we put the module that we have in the input data.

 

We are going to perform a static calculation, where we will apply the imposed displacement and we will associate it to a modal calculation, which will allow us to calculate the modality taking into account this introduced initial tension associated with the imposed displacement.

Figure 1. Calculations to be done

Let's start by seeing what the frequency of vibration of a string is without any imposed displacement, that is, an initial situation with a displacement of 0mm.

 

Figure 2. First mode of vibration. Vibration frequency for the initial situation (0mm of displacement) 

If you look at the top, the vibration frequency of the first mode is 4.4Hz.

Now the question is - do we loose or stretch the rope? We will stretch it. When we introduce a tension in a cable what we are doing is adding a term to the corresponding stiffness matrix; this additional term is called "geometric rigidity". Similarly we can think of the buckling produced by a compression that introduces a geometric term which is deducted in the matrix of rigidity leading to a loss of rigidity.

 

To obtain the requested frequency we will test with 1mm of displacement. We do the static calculation, we apply the modal with the "pre-stress" and we see the vibration modes. In the following image we see that for 1mm of displacement we obtain 176Hz. It's not bad ... but it's not the solution.

 

Figure 3. First mode of vibration. Vibration frequency for the 1mm displacement situation

Keeping the stretching idea, we will give another lug of 1mm to the string such that we have stretched a total of 2mm. We look at the displacement results that we are actually giving this displacement and not another (we check that 2mm are displaced at the end).

 

Figure 4. Imposed displacement (2mm)

And doing so, in the same way, its frequency of vibration associated with the first mode is 249HZ.

Figure 5. First mode of vibration. Vibration frequency for the 2mm displacement situation

It's a matter of trying a bit more and we get the value. We summarize the results:

0mm –> 7.4 Hz

1mm –>176 Hz

2mm –> 249

4mm –> 353 Hz

6mm –> 431Hz

6.5mm –> 449.5Hz

We accept 449.5Hz close enough to 450 Hz and we end up with 6.5mm. Well, we already know how much to stretch the string so that it vibrates at 450 Hz!

We are going to do one last thing to check the results. We calculate the force that would be necessary to apply to cause these 6.5mm of displacement, we include it in Ansys and we verify at what frequency the string vibrates with the force applied.

We apply these 446.69 N on the string and see what happens. Very well! We see that the rope stretches the 6.5mm that we expected and that also vibrates at 449.6Hz.

Figure 6. Displacement of the string for 44669N tension force

Figure 7. Frecuency of the first mode for the 44669N tension force case

 

We finish showing the other vibration modes of the string and their associated frequencies:

Figure 8. Frecuency of the other modes for the 44669N tension force case

 

I hope you liked this simple explanation about the vibration of a guitar string and how applying displacements at its end we can vary its frequency.

As a curiosity I end up saying that sometimes to ensure that the tendons of the bridges have been properly tensioned, the cable is vibrated and it is verified that it vibrates at the frequency associated with the tension to which we wanted to tension it.

Resultado de imagen de tesado de un cable de puente

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About the author
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Carlos Corral . MEng Civil Engineering from the Politécnica university of Madrid. Speciality: Structural engineer. Owner and programer of Prontubeam.com and Prontubeam.com/en.
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